Nonlinear Combustion Instability in

Liquid-Propellant Rocket Motors

Samuel Z. Burstein and Wallace Chinitz

Second Quarterly Report

to . -

Jet Propulsion Laboratory

January 30, 1968

JPL COMTBACT 951946

"This work was performed for the Jet Propulsion Laboratory, California Institute of Technology, as sponsored by the National Aeronautics and Space Administration under Contract NAS7-100."

https://ntrs.nasa.gov/search.jsp?R=19680009979 2018-08-29T01:57:03+00:00Z

"This report contains information prepared

by Mathematical Applications Group, Inc.

(MAGI) under JPL subcontract. Its content

is not necessarily endorsed by the Jet

Propulsion Laboratory, California Institute

of Technology, or the National Aeronautics

and Space Administration."

i

TABLE OF CONTENTS

INTRODUCTION ...................................... 1

DROPLET EVAPORATION AND COMBUSTION ANALYSES ....... 2

A . The Rate of the N2H4/N204 Reaction ........... 2 B . The Diffusion-Controlled Flame Analysis ..... 11

. FLUID DYNAMIC MODEL .............................. 17

A . First-Row Mesh Point Calculation ............ 17 B . An Exact Solution-Comparison with

Machine Experiment ......................... 19

C . A Finite Amplitude Calculation of a Rotating Wave .............................. 21

D . Convergence of the Difference Equations ..... 2 3

E . Simple Forcing Function ..................... 2 6

F . Program COMB Status ......................... 28

ii

INTRODUCTION

During the second quarter of the contract there has been

significant progress in the construction of a mathematical model

describing the dynamics of a rotating finite-amplitude wave in a

cylindrical combustion chamber. This quarterly report describes

some of the details of this activity.

The first section describes the theory of finite rate proc-

esses in the combustion of hydrazine (NzHq)/nitrogen tetroxide

(N2O4);

for the modified surface flame model of combustion. Also discussed

the results will be used to predict kinetic parameters

in this section are some results obtained from the diffusion flame

analysis.

The second section is involved with the fluid dynamic model.

A new treatment for the calculation of special mesh points in

COMB is presented. Most important, however, the results of some

time dependent computations are described in which a finite am-

plitude pressure wave is allowed to rotate and steepen in a

cylindrical chamber. The reader should refer to reference 1 for

background material and additional references (especially reference

2 0 ) .

The authors would like to acknowledge the significant contri-

bution of Harold Schechter in the organization, analysis and

programming of the digital programs being developed in this

research program.

1

I. THE DROPLET EVAPORATION AND COMBUSTION ANALYSES

A . The Rate of the N2H4/N204 Reaction

It will be recalled (Ref. 1) that the species conservation

equation including finite rate chemical reaction can be written:

dYi - d 2 dYi 2 N F yNo (1) a- ( n -) = - iin bYF dn 0

where a = the dimensionless mass burning rate

Yi = mass fraction of species i

n = radius/droplet radius

Mi = molecular weight of species i

p = mass density

NF,N, = reaction orders 2

b = rDk/D

k = Arrhenius reaction rate constant = A(T)exp(-E/RT)

D = diffusion coefficient

It was also shown (Ref. 1) that by neglecting the energy

transport due to concentration gradients (the Dufour effect) and

assuming the Lewis number to be unity, the energy equation takes

on the same form as equation (1) above:

2 NF NO a- - - - i q b Y F (,,* dt -

dn F

dt dn dn -1 -

2

In order to apply Peskin's (Ref. 2) modified flame surface

analysis, realistic estimates must be obtained for the reaction

rate constant, k = A (T) exp (-E/RT) , and the reaction orders, NF and No, corresponding to the N2H*/N204 reaction. These can

be obtained from a one-dimensional, finite-rate analysis of the

oxidation process, using the simplified chemical kinetic mechanism

deduced from Sawyer, and discussed in Ref. 1. This method is

formulated in the following manner.

As mentioned above, the flow equations are written for one-

dimensional, constant pressure, inviscid flow along streamtubes,

with the

a.

b.

C.

following additional assumptions:

Negligible diffusion normal to the flow direction,

leading to a uniform mixture of (gaseous) species

at any streamwise point. Therefore, the oxidation

is taken to be 'reaction-controlled'.

The flow is steady and adiabatic.

The initial concentrations of the oxidizing species

are obtained in accordance with the following

reaction equation and with the assumption of

chemical equilibrium:

N2O4 + a N204+bN02 + cNO + d02 ( 3 )

The method for obtaining the equilibrium species concentrations

is detailed in Appendix A.

With these assumptions, the relevant equations are:

Energy Conservation

dh = ; h = constant dt

(4)

where h includes sensible - and chemical enthalpy.

3

Species Conservation

dYi = ri Pdt (5)

Equation of State

P = p M RT

Auxiliary Equations

h = C Yihi i

M = C Yi (i F)

The enthalpy terms are taken to be linear in form: -

hi = Ai + c Pi

where the reference temperature was chosen to be 1000'K. Values

for Ai and E

chemical kinetic mechanism and t h e species generation terms, ri,

are in Tables I and 11, respectively. The reaction rate constants

are in Table I.

were obtained from the data of Reference 5. The Pi

It should be emphasized that this premixed, homogeneous gas-

phase analysis is designed to supply an appropriate representation

of the overall (global) oxidation reaction rate, for use in equa-

tions (1) and ( 2 ) . It is not, of course, a representation of the

processes occurring around the fuel drop, where a counter-flow

diffusion flame exists, as described by equations (1) and ( 2 ) .

Results for two values of the ambient pressure are shown in

Figures 1 and 2 . At one atmosphere (Figure 11, and an initial

temperature of 1500'K, the reaction is seen to commence at about

seconds, continue smoothly until about seconds after

4

b

. which a rapid rise in temperature is noted. If the calculations

were continued, a rapid leveling off to the adiabatic flame tem-

perature would be noted. In this case, the ignition delay time

can be estimated to be seconds.

At the lower initial temperature (Ti = 1000°K), the ignition

delay time is seen to increase in Figure 1 to about S X ~ O - ~ seconds.

When the pressure is increased to 20 atmospheres (Figure 2 ) ,

a somewhat different behavior is noted. At both initial temper-

atures, the reaction is seen to commence quite early (at about

10’’ seconds for Ti = 1500°K, and before lo’’ seconds for Ti =

1000°K). However, a pause in the reaction then occurs and, for

the case of Ti = 1500°K, between 5x10-’ and 5x10-* seconds little

temperature change is noted. After 5x10-* seconds, a smooth and,

ultimately, rapid rise in temperature is noted.

The reason for this behavior can be explained by reference

to Figure 3 and Table 11. In Figure 3, it is seen that NO2 dis-

appears rapidly during this initial temperature rise, with a

corresponding decrease in the concentration of hydrazine. This

same behavior was not noted f o r the case of p = 1 atm because

reaction number 2 in Table I1 is second-order and, hence, pre-

dominates at higher pressures. An accompanying increase of NO,

as a result of reaction 2, can be seen in Figure 3 as well.

After the depletion of the N02, the N2H4/NO and N2H4/O2

oxidation reactions take over and proceed as first-order reactions

(Table 11), leading to a second ‘delay’ period, and a subsequent

rapid reaction at about seconds. The behavior of NH3 and

6

H 2 is shown in Figure 4 . It can be seen that they rise in concen-

tration initially, as a consequence of the hydrazine decomposition

reaction (reaction 1, Table 11), then rapidly disappear when the

main reaction occurs.

It is clear then, that the 'ignition delay time' is difficult

to define in the pressure regime where this two-step reaction

occurs. It can be anticipated that the traditional plot of igni-

tion delay as a function of reciprocal temperature will not produce

a straight line since the rate-controlling reactions are different

at different pressures. This will lead to the requirement for a

different 'global' reaction rate constant for the high and low

pressure regimes. Current efforts are directed toward obtaining

sufficient computer runs so that the chemical kinetic parameters

can be determined in both pressure regimes.

9

0 l-i

U I

B. The Diffusion-Controlled Flame Analysis

The diffusion-controlled flame analysis which was described

in Reference 1 has been coupled to the basic computer program

COMB. In this section, we compare the results obtained from this

analysis with those obtained from a more elaborate analysis which

has been published in the literature, as well with some experi-

mental results.

It will be recalled that the modified Godsave equation de-

veloped in Reference l was 1

where 61;

dL

k

P C

Tm

TL

Y o p

4

f S

AHR

LF

= mass burning rate in stagnant surroundings

= droplet diameter

= fuel thermal conductivity

= fuel specific heat

= ambient temperature

= saturation temperature of the fuel

droplet at the chamber pressure

= ambient mass fraction of oxidizer

= fuel-oxidant equivalence ratio

= stoichiometric fuel-oxidant ratio

= heat of reaction of the mixture

= latent heat of the fuel

Figure 5 shows the results of the solution of equation ( 3 ) in

terms of the dimensionless burning rate, W G fiF/(2ndLk/cp), for 0

11

c r

two values of equivalence ratio and 300 psia chamber pressure.

The curves of W are virtually straight lines, and are nearly

independent of T,. These results are, of course, for the

N 2 H 4 / N 2 0 4 system.

Also shown are the theoretical results of Reference 3.

Strictly speaking, these results are not directly comparable with

those presented herein since they are apparently for hydrazine

decomposition (in the absence of N 2 0 4 ) and, probably, at one

atmosphere. Of principle interest, however, are the facts that

the orders-of-magnitude are the same, but that the results of

Reference 3 indicate a much greater dependence of W on the ambient

temperature. This stems principally from the fact that the analy-

sis of Reference 3 includes a finite-rate reaction term (taken to

be first-order) which is sensitive to the ambient temperature.

The modified f lame-sheet analysis being developed under this pro-

gram contains a finite-rate reaction term as well, and can be

expected to show a like dependence on T,.

Perhaps a more apt comparison is that shown in Figure 6 .

The results obtained from equation (3) are compared with some

experimental results obtained from Reference 4. These latter re-

sults are presumably at one atmosphere, but the ambient temperature

is not specified in Reference 4. As can be seen, equation ( 3 )

predicts the mass burning rate reasonably well and, of course,

predicts the linear variation of fiz with dL. l

The effects of the three variables T,, p,, and I$ on W are

summarized in Figure 7. As can be seen, (a) reducing the equivalence

13

.

?atio leads to the expected reduction in W, (b ) increasing pressure

leads to an increase in W, but a reduction in mF (Fig. 61, at

least at an ambient temperature of 1 0 3 5 O K (this result will not

occur f o r all values of T m ) , and (c) increasing T, leads to an

increase in W.

.o

16

11,. FLUID DYNAMIC MODEL

A . First Row Mesh Point Calculation

In this section we describe a new method of computing the

solution to the partial differential equations of a compressible

fluid in cylindrical coordinates in the neighborhood of the

singular point r = 0 . In Reference 1, the mesh point at r = o

was computed with the help of contour integrals on the circle of

radius Ar (the mesh considered is one in which mesh points are

defined by the intersection of curves r = constant and e = con-

stant). All mesh points on r = Ar (Ar is the mesh space in the

radial direction) cannot be computed by the standard method now

being used for interior mesh points because first row points do

not have eight nearest neighbors. The following technique is used

to define the required eight nearest neighbors.

The method consists in the construction of two temporary sets

of data points at r = Ar/2 and r = 3~r/2. These points are ob-

tained by interpolation formulae from information known at r = 0 ,

Ar and 2Ar. Higher order formulae would require data at additional

points ..., -Ar, ..., 3Ar, .... A submesh is then constructed in

the neighborhood of the ring of radius Ar in which the mesh spacing

in the tangential direction is the same as before but the radial

mesh size is reduced by a factor of two. The calculation then

proceeds as if the set of mesh points with coordinates (Ar, e j ,

j = 1, ..., J M A X ) are interior mesh points. The subroutine which

is responsible for this calculation (FRSTRO) has been rewritten

1 7

to incorporate the new method.

routine GENPT as a result of the creation of the new mesh points

This routine makes use of sub-

(nr/2, $ j ) and (3Ar/2, $j) on the two respective circles and

eliminates some special subroutines that existed in COMB originally.

18

‘B. An Exact Solution-Comparison with Machine Experiment

We have considered the exact solution (linearized solution)

to the differential equation in the potential 0

The motion is irrotational so that the velocity in the ith direction

ui is the gradient of the potential + . In addition the require-

ment that the normal velocity vanish on the boundary S of the Xi

chamber

We find that

and

The velocity components are

-1 €(Kr)J1(Kr) cos (K

Figures (8) and (9) give the density field (pl/Y) and the pressure

field at the start of the computation, p = p(r,$,O). This is a

qraphical representation of the linearized solution. The above

velocity components go into the construction of the velocity field

which is shown in Figure (12); however, this plot corresponds to

the finite amplitude case which will be discussed later. The

19

velocity plots scale, the maximum and minimum value for the in-

finitesimal case is 10 those values shown in Figure (12).

Figure (10) shows the pressure field after 6 0 0 cycles of calcula-

tion; the non-dimensional time at this point is 5 .69 (x.131

millisec - .746 millisec real time) and the wave has undergone

1 . 6 2 revolutions or approximately 2,200 rev/sec (period = 460

u-sec). Figure (11) shows the corresponding field obtained from

the exact solution - the results are virtually identical; if the

two plots are superimposed, the point of maximum pressure is

congruent.

-5

The solution will, if calculations are allowed to continue,

appear to undergo uniform rotation at the given period. No dis-

tortions or asymmetries appear. Nonlinear effects, the bunching

of the isobars in the compression portion of the field, are absent.

This characteristic property is absent in the next problem con-

sidered.

20

C. A Finite Amplitude Calculation of a Rotating Wave

Only one parameter has been changed in the previous calcula-

tion; that parameter is the maximum pressure (450 psia). The

velocity field at t = o corresponding to this pressure is given

in Figure (12). Figures (12)-(18) show in detail, the motion of

the velocity field up to 600 cycles of computation (corresponding

to a value of non-dimensional time of 1.857 x 2.18 = 4 . 0 5 ; all

time values for the finite amplitude case should be multiplied

by this factor 2.18, i.e., Figures ( 1 2 ) - ( 3 6 ) ) . From Figures (18)

to (21) a plot of the velocity field each 600 cycles of computation

is shown. It is clear that

a) the velocity field does not remain symmetric;

b) an induced flow is generated behind the

compression wave;

c) this flow is fully developed in about one

period of rotation.

Figure (21) was plotted poorly due to the Calcomp having a stuck

pen; the actual velocity is not as large as indicated at the

chamber walls.

Figures ( 2 2 ) to ( 3 1 ) are the contours of the density field

corresponding to the previous figures which represented the velocity

field. Figure ( 2 2 ) is the initial field at t =o. It is clear, by

scanning through these figures that one observes that

a) the density field steepens in the leading edge

of the compression pulse and spreads in the

region of the trailing edge;

21

b) this steepening is most pronounced in the first

half to three quarters of the first period;

c) a periodic (wt+2n) continuous wave solution seems

to be the asymptotic solution (t-). A total of

about 4 3/4 rotations were computed and shown in

these figures.

These observed effects are clearly nonlinear in nature.

Figures (32) to (36) are the plots of the pressure field at

intervals of 600 cycles. The initial pressure distribution is

shown in Figure (32). The pressure pulse steepens up to a maximum

in about one period and then is approximately constant -1.6 (575

psia).

22

D. Convergence of the Difference Equations

It is to be expected that as the mesh gets finer and finer,

the solution will converge to the differential equation.

truncation error for the approximation used in this calculation

is second order every where (except for the evaluation of the

interpolated data for the first row calculation).

step decreases the dissipation of the scheme should decrease,

i.e., finer meshes should lead to sharper results. We have tried

two meshes, the finer one (all the results presented have been

computed on this mesh), 10 points in r and 3 6 points in 6 , and

the coarse one, 5 points in r and 18 points in 6. The figures

( 3 7 ) , ( 3 8 ) and ( 3 9 ) give the value of the density,pressure and

velocity at a non-dimensional time of 1 4 . 7 for the coarse mesh.

The

As the space

These results may be compared to the fine mesh solution at 2400

cycles which, in turn, corresponds to a non-dimensional time of

16.0. It is clear that much of the detail of the solution is

absent in the formal calculation. The two tables which follow

give some additional means for comparing the two solutions.

2 3

5 x 18 MESH

I 600 14.7 1.244 0.414 0.521 1.37 0.665

540 13.3 1.35 0.393 0.559 1.47 0.709 I 480 11.8 1.49 0.379 0.586 1.49 0.744

420 10.4 1.45 0.369 0.571 1.58 0.721

360 8.9 1.42 0.397 0.502 1.46 0.651

300 7.4 1.21 0.378 0.505 1.44 0.654

240 5.9 1.44 0.376 0.517 1.49 0.669

180 4.4 1.57 0.342 0 . 5 8 1 1.59 0.740

120 3.0 1.55 0.409 0.557 1.62 0.716 I I 60 1.6 1.35 0.393 0.533 1.45 0.691

0 0.0 1.25 0.358 0.417 1.40 0.561

24

,

1 0 X 36 MESH

+ Pmax Urnax Pmin P m a x Pmin C y c l e t/to

2400

1 8 0 0

1 2 0 0

6 0 0

5 4 0

420

300

1 8 0 I

16 .0

1 2 . 5

7.8

4.05

3.66

2.92

2 .14

1 . 3 1

1 . 6 7

1 . 5 9

1 . 5 1

1 . 8 3

0 .434

0 .440

0 .385

0.448

0.454

0.486

0 .399

0 .472

0 .547

0 .542

0 .509

0 .562

1 . 7 4

1 . 6 7

1 . 6 1

1 .87

1 .94

1 .70

1 .73

1 .50

0 .704

0 . 7 0 1

0 .666

0 .715

0 .729

0 .720

0 .655

0 . 6 6 9

I 6 0 0.45 -- 0 .395 --- 1 .49 0.645 I

0 0 . 0 1 . 2 5 0.358 0.417 1 .40 0 . 5 6 1

E. Simple Forcins Function

Since the non-reacting code is yielding fairly accurate solu-

tions a first attempt at calculating the flow resulting from the

differential equation

W t + F r + G q + S + H , = $ , J, =(i) where the forcing function $, which has a single component E is

given by

subject to the initial data and boundary conditions of the problem

described in Section C.

At t = 0, the energy is switched on. Since the energy release

is biassed towards high pressures the most intense region of com-

bustion occurs about @=O near the boundary (where r=R). A very

strong wave, turning into a shock, is propagating in towards the

chamber center almost symmetric about 8=0. Figures (40) and (41)

show the pressure field and velocity field after 10.5 psec

(= .0376~2.18~.131*10-~) . The pressure reaches approximately

20,600 psia, while the maximum value of the velocity is approxi-

mately 10,000 ft./sec. (2.87~3500). The harshness of these results

indicate the intensity of the energy source. This calculation

became unstable at the center after approximately 30 cycles. The

shock has not yet reached this point. This calculation is to be

26

considered only as an exploratory effort and it is interesting

only in that the severe gradients generated behave well, i.e.,

the difference equations seem stable in the region which contains

a strong shock.

27

F. Program COMB Status

The forcing function is being changed in program COMB. There

are two modifications in progress.

function described in Section E. Rather than switching on the

energy source immediately, we plan to allow the calculation to

run several tens of cycles (to allow equilibrium of the data on

the discrete mesh to occur) and then turn on this source, not at

once but rather slowly.

One concerns using the simple

Perhaps it may require a,full rotation

of the finite amplitude wave around the chamber before the energy

is completely turned on.

The second change will allow the coupling of the actual chem-

ical and droplet (which has been discussed in the first two sections

of the report) model to the gas dynamic model.

In addition,. a still finer ‘mesh (on which the numerical solu-

tion is obtained) is to be used to compare the convergence of the

solution with the two calculations reported in Section D. This

mesh will contain 72x20 mesh points.

28

29

30

31

.L(1 -.2U . . l o - .oo o.oooppoo 0 eo 30

' 4 1 -.Y1 - -31 - .ZU - . I C - .uo .IO .203J'j70 60

VECTOP PLBl OF UMJIPG VECTOR -50 [NCHES LePrG = 4.72E-01. VRLUES S 5.67E-02 RRE e 0 6 [NCHES LONG.

- .41 - .31 -.IC - .l!O 10 ‘30 .Y1 .3843%6Lt4 300

- .31 - .211 -.in ~ .LlU .IO 1 .3Ltjqlll Lt20

. i0

VECTOR PLOT 13F UMAG VECTOR *SO tNCHES LONG 1 Y-YBE-131- VRLUF.5 S 5.37E-02 ARE .[E LNCHES LCNGZ

VECTOR PLOT OF UMQG - VECTOR .50 [NCHES LDNG I 9.9OE-01. VRLES 5 5.28E-02 RRE -06 INCHES LDNG.

'-.SI -.Y1 -.a - . l e - .oo I1C .20 -30 .51 5.57Y397P 1800

4 2

4 3

4 4

I t L 1

f

i t I i

I

i i F

f i I

t !

T 4 I

i i I

1 I

-i !

4

I i I -! , j

I

i

I

j 1 I

j

46

4 9

.

5 0

51

52

9

5 6

.

57

59

.

61

62

REFERENCES

1. Burstein, S. Z. and Chinitz, W., "Nonlinear Combustion Instability in Liquid Propellant Rocket Motors", First Quarterly Report, Mathematical Applications Group, Inc., October 30, 1967.

2. Peskin, R. L. and Wise, H., AIAA Journal, 4, 1646 (1966). -

3. Dynamic Science Quarterly Progress Report, "Combustion Instability and Random Wave Phenomena", Contract NAS7- 467, SN-95, October 31, 1967.

4. Ibid, Monthly Status Report SN-95-4.

5. Dow Chemical Company, JANAF Thermochemical Tables.

63

APPENDIX A

Calculation of Nitroaen Tetroxide Eauilibrium

Nitrogen tetroxide (N2O4) is assumed to decompose according

to the following reaction equation:

N204 -f aN2O4+bNO2+cN0+dO2

Atomic conservation yields:

N: 2 = 2 a + b + c

0: 4 = 4 a + 2 b + c + 2 d

The equilibrium equations selected were: .- 2

P

4 +

N2O4 = 2N02: K1= K2 - x1

NO2 = N0+%02: K2= x3x4p x2 (A. 5 )

where Xi is the mole fraction of species i (1 = N204, 2 = NO2,

3 = NO, 4 = 02), Kj is the equilibrium constant for reaction j ,

and p is the pressure.

Equations (A.2) - (A.5) are sufficient to solve for the four unknown species concentrations once p and T are specified. (It

will be recalled that K, = Kj (T) ,

.) The Newton-Raphson iterative

this system of equations.

and are obtained from Reference

technique was employed to solve

The calculations yield the anticipated result that the con-

centration of N2O4 over most of the range of likely engine operating

conditions is negligible. This accounts for its neglect in the

chemical kinetic mechanism of Table 11.

64

t

0 12 4 B W m -lz

0 0 12B

W N

at%

0 0 W B

m m -lz

0 0 N B 12 m

c 0 -4 4J 0 m 2 0 z

-lz

0 0 COB

m m - &

N rl d (Y I N N

0 0 12B

03 m

0 X + 0 z W f 0 z + X z

N

hl

* d

N

alz

n N

0 0 O B 00 In

N z N + 0 X f 0 z N + X z

N

N

d

N

-lz

n m

0 0 N B 03 m

0 cv

alz

N X z rnlN

N Z x + N N O 0 z X m N X z z

6 8 N

+ m q d

H

B

n n Q In

0 1 :

n W

0 N

FIN

N z X + N

0- m k + m

X z

n

12

0 N

X + 0 z + 0 z X

N

+N

n 03

n

c a 0 X -4 a,

u \ o ( d m a, g o

rl 0 0 z d

I rl

N z X

8

+N

(v X f 0 z X

n m

0 X + 0 X + X

(v

N

N

n 0 rl

4 TABLE I1

SPECIES GENERATION EQUATIONS

(e) R7 = p Y g M 7 + 7 k5 p Y 2 + k8 pY3 - 2k3 p Y 1

M6 M2 M3 M1 - - z

(9) Rg = 2 p Y 1 3 (,k2 - Py6 + k3 + k4) + 3 P Y 2 M9 M1 M6 7 M21

6 6